4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson...

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142 Chapter 4 Graphing and Writing Linear Equations Graphing Linear Equations 4.1 How can you recognize a linear equation? How can you draw its graph? Work with a partner. a. Use the equation y = 1 2 x + 1 to complete the table. (Choose any two x-values and find the y-values.) b. Write the two ordered pairs given by the table. These are called solution points of the equation. c. PRECISION Plot the two solution points. Draw a line exactly through the two points. d. Find a different point on the line. Check that this point is a solution point of the equation y = 1 2 x + 1. e. LOGIC Do you think it is true that any point on the line is a solution point of the equation y = 1 2 x + 1? Explain. f. Choose five additional x-values for the table. (Choose positive and negative x-values.) Plot the five corresponding solution points. Does each point lie on the line? Solution Points x y = 1 2 x + 1 g. LOGIC Do you think it is true that any solution point of the equation y = 1 2 x + 1 is a point on the line? Explain. h. Why do you think y = ax + b is called a linear equation? ACTIVITY: Graphing a Linear Equation 1 1 Solution Points x y = 1 2 x + 1 x y 3 4 5 6 2 1 3 4 5 6 2 4 5 6 3 2 1 O 2 3 4 5 6 Graphing Equations In this lesson, you will understand that lines represent solutions of linear equations. graph linear equations.

Transcript of 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson...

Page 1: 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson Lesson Tutorials Key Vocabulary linear equation, p. 144 solution of a linear equation,

142 Chapter 4 Graphing and Writing Linear Equations

Graphing Linear Equations4.1

How can you recognize a linear equation?

How can you draw its graph?

Work with a partner.

a. Use the equation y = 1

— 2

x + 1 to

complete the table. (Choose any two x-values and fi nd the y-values.)

b. Write the two ordered pairs given by the table. These are called solution points of the equation.

c. PRECISION Plot the two solution points. Draw a line exactly through the two points.

d. Find a different point on the line. Check that this point is a solution

point of the equation y = 1

— 2

x + 1.

e. LOGIC Do you think it is true that any point on the line is a solution

point of the equation y = 1

— 2

x + 1? Explain.

f. Choose fi ve additional x-values for the table. (Choose positive and negative x-values.) Plot the fi ve corresponding solution points. Does each point lie on the line?

Solution Points

x

y = 1 —

2 x + 1

g. LOGIC Do you think it is true that any solution point of the equation

y = 1

— 2

x + 1 is a point on the line? Explain.

h. Why do you think y = ax + b is called a linear equation?

ACTIVITY: Graphing a Linear Equation11

Solution Points

x

y = 1 —

2 x + 1

x

y

3

4

5

6

2

1

3

4

5

6

2

4 5 6321O23456

Graphing Equations In this lesson, you will● understand that lines

represent solutions of linear equations.

● graph linear equations.

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Page 2: 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson Lesson Tutorials Key Vocabulary linear equation, p. 144 solution of a linear equation,

Section 4.1 Graphing Linear Equations 143

Use a graphing calculator to graph y = 2x + 5.

a. Enter the equation y = 2x + 5 into your calculator.

b. Check the settings of the viewing window. The boundaries of the graph are set by the minimum and the maximum x- and y-values. The numbers of units between the tick marks are set by the x- and y-scales.

c. Graph y = 2x + 5 on your calculator.

d. Change the settings of the viewing window to match those shown.

Compare the two graphs.

ACTIVITY: Using a Graphing Calculator22

Use what you learned about graphing linear equations to complete Exercises 3 and 4 on page 146.

3. IN YOUR OWN WORDS How can you recognize a linear equation? How can you draw its graph? Write an equation that is linear. Write an equation that is not linear.

4. Use a graphing calculator to graph y = 5x − 12 in the standard viewing window.

a. Can you tell where the line crosses the x-axis? Can you tell where the line crosses the y-axis?

b. How can you adjust the viewing window so that you can determine where the line crosses the x- and y-axes?

5. CHOOSE TOOLS You want to graph y = 2.5x − 3.8. Would you graph it by hand or by using a graphing calculator? Why?

This is thestandard viewingwindow.

10 10

10

10

y 2x 5

6 2

4

8

y 2x 5

Recognize Usefulness of ToolsWhat are some advantages and disadvantages of using a graphing calculator to graph a linear equation?

Math Practice

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Page 3: 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson Lesson Tutorials Key Vocabulary linear equation, p. 144 solution of a linear equation,

144 Chapter 4 Graphing and Writing Linear Equations

Lesson4.1Lesson Tutorials

Key Vocabularylinear equation, p. 144solution of a linear equation, p. 144

Linear Equations

A linear equation is an equation whose graph is a line. The points on the line are solutions of the equation.

You can use a graph to show the solutions of a linear equation. The graph below represents the equation y = x + 1.

RememberAn ordered pair (x, y ) is used to locate a point in a coordinate plane.

x y (x, y)

−1 0 (−1, 0)

0 1 (0, 1)

2 3 (2, 3)

EXAMPLE Graphing a Linear Equation11

Graph y = −2x + 1.

Step 1: Make a table of values.

Step 2: Plot the ordered pairs.

Step 3: Draw a line through the points.

x

y

1O23 2 3

1

2

3

3

2

(0, 1)( 1, 0)

(2, 3)

y x 1

x

y

23 2 3 4

2

1

3

3

( 1, 3)

(2, 3)

(0, 1)y 2x 1

O

Graphing Horizontal and Vertical Lines

The graph of y = b is a horizontal line passing through (0, b).

The graph of x = a is a vertical line passing through (a, 0).

y

x

(a, 0)

x a

O

y

x

(0, b)

y b

O

x y = −2x + 1 y (x, y)

−1 y = −2(−1) + 1 3 (−1, 3)

0 y = −2(0) + 1 1 (0, 1)

2 y = −2(2) + 1 −3 (2, −3)

Check

43

3

3

y 2x 1

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Page 4: 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson Lesson Tutorials Key Vocabulary linear equation, p. 144 solution of a linear equation,

Section 4.1 Graphing Linear Equations 145

EXAMPLE Graphing a Horizontal Line and a Vertical Line22

a. Graph y = −3. b. Graph x = 2.

The graph of y = −3 is a horizontal line passing through (0, − 3). Draw a horizontal line through this point.

Graph the linear equation. Use a graphing calculator to check your graph, if possible.

1. y = 3x 2. y = − 1

— 2

x + 2 3. x = − 4 4. y = −1.5

EXAMPLE Real-Life Application33The wind speed y (in miles per hour) of a tropical storm is y = 2x + 66, where x is the number of hours after the storm enters the Gulf of Mexico.

a. Graph the equation.

b. When does the storm become a hurricane?

a. Make a table of values.

x y = 2x + 66 y (x, y)

0 y = 2(0) + 66 66 (0, 66)

1 y = 2(1) + 66 68 (1, 68)

2 y = 2(2) + 66 70 (2, 70)

3 y = 2(3) + 66 72 (3, 72)

Plot the ordered pairs and draw a line through the points.

b. From the graph, you can see that y = 74 when x = 4. So, the storm becomes a hurricane 4 hours after it enters the Gulf of Mexico.

5. WHAT IF? The wind speed of the storm is y = 1.5x + 62. When does the storm become a hurricane?

A tropical storm becomes a hurricane when wind speeds are at least 74 miles per hour.

x

y

1 2 3 4 5 6

60

64

68

72

76

y 2x 66

Exercises 5–16

The graph of x = 2 is a vertical line passing through (2, 0). Draw a vertical line through this point.

x

y

1O23 2 3

2

4

(0, 3)

y 3

x

y

1O2 3 4

2

1

2

(2, 0)

x 2

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Page 5: 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson Lesson Tutorials Key Vocabulary linear equation, p. 144 solution of a linear equation,

Exercises4.1

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

146 Chapter 4 Graphing and Writing Linear Equations

1. VOCABULARY What type of graph represents the solutions of the equation y = 2x + 4?

2. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three? Explain your reasoning.

y = 0.5x − 0.2

4x + 3 = y

y = x 2 + 6

3

— 4

x + 1

— 3

= y

PRECISION Copy and complete the table. Plot the two solution points and draw a line exactly through the two points. Find a different solution point on the line.

3. x

y = 3x − 1

4. x

y = 1 —

3 x + 2

Graph the linear equation. Use a graphing calculator to check your graph, if possible.

5. y = −5x 6. y = 1

— 4

x 7. y = 5 8. x = − 6

9. y = x − 3 10. y = −7x − 1 11. y = − x

— 3

+ 4 12. y = 3

— 4

x − 1

— 2

13. y = − 2

— 3

14. y = 6.75 15. x = − 0.5 16. x = 1

— 4

17. ERROR ANALYSIS Describe and correct the error in graphing the equation.

18. MESSAGING You sign up for an unlimited text-messaging plan for your cell phone. The equation y = 20 represents the cost y (in dollars) for sending x text messages. Graph the equation. What does the graph tell you?

19. MAIL The equation y = 2x + 3 represents the cost y (in dollars) of mailing a package that weighs x pounds.

a. Graph the equation.

b. Use the graph to estimate how much it costs to mail the package.

c. Use the equation to fi nd exactly how much it costs to mail the package.

Help with Homework

11 22

x

y

O

1

2

3

4

1 2 3

y 4

(4, 0)

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Page 6: 4.1 Graphing Linear Equations · 144 Chapter 4 Graphing and Writing Linear Equations 4.1 Lesson Lesson Tutorials Key Vocabulary linear equation, p. 144 solution of a linear equation,

Section 4.1 Graphing Linear Equations 147

Write the ordered pair corresponding to the point. (Skills Review Handbook)

28. point A 29. point B

30. point C 31. point D

32. MULTIPLE CHOICE A debate team has 15 female members. The ratio of females to males is 3 : 2. How many males are on the debate team? (Skills Review Handbook)

○A 6 ○B 10 ○C 22 ○D 25

Solve for y. Then graph the equation. Use a graphing calculator to check your graph.

20. y − 3x = 1 21. 5x + 2y = 4

22. − 1

— 3

y + 4x = 3 23. x + 0.5y = 1.5

24. SAVINGS You have $100 in your savings account and plan to deposit $12.50 each month.

a. Graph a linear equation that represents the balance in your account.

b. How many months will it take you to save enough money to buy 10 acres of land on Mars?

25. GEOMETRY The sum S of the interior angle measures of a polygon with n sides is S = (n − 2) ⋅ 180°.

a. Plot four points (n, S ) that satisfy the equation. Is the equation a linear equation? Explain your reasoning.

b. Does the value n = 3.5 make sense in the context of the problem? Explain your reasoning.

26. SEA LEVEL Along the U.S. Atlantic coast, the sea level is rising about 2 millimeters per year. How many millimeters has sea level risen since you were born? How do you know? Use a linear equation and a graph to justify your answer.

27. ProblemSolvingProblemSolving One second of video on your digital camera uses the same

amount of memory as two pictures. Your camera can store 250 pictures.

a. Write and graph a linear equation that represents the number y of pictures your camera can store when you take x seconds of video.

b. How many pictures can your camera store in addition to the video shown?

Video time:1 min. 30 sec.

A

y

x2246 4

B

CD

6

4

2

2

4

O

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